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| Description: Absorption of existence condition by uniqueness. (Contributed by NM, 4-Nov-2002.) Shorten proof and avoid df-eu 2568. (Revised by BJ, 14-Oct-2022.) | 
| Ref | Expression | 
|---|---|
| moabs | ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-1 6 | . 2 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃*𝑥𝜑)) | |
| 2 | nexmo 2540 | . . 3 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) | |
| 3 | id 22 | . . 3 ⊢ (∃*𝑥𝜑 → ∃*𝑥𝜑) | |
| 4 | 2, 3 | ja 186 | . 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) → ∃*𝑥𝜑) | 
| 5 | 1, 4 | impbii 209 | 1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∃wex 1778 ∃*wmo 2537 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 | 
| This theorem depends on definitions: df-bi 207 df-ex 1779 df-mo 2539 | 
| This theorem is referenced by: mo3 2563 mo4 2565 moeu 2582 dffun7 6592 wl-mo3t 37578 | 
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